Complexity of the usual torus action on Kazhdan–Lusztig varieties

Donten-Bury, Maria; Escobar, Laura; Portakal, Irem

doi:10.5802/alco.279

Donten-Bury, Maria ¹; Escobar, Laura ²; Portakal, Irem ³

¹ University of Warsaw Institute of Mathematics Banacha 2 02-097 Warszawa Poland
² Department of Mathematics and Statistics Washington University in St. Louis One Brookings Drive St. Louis, Missouri 63130 U.S.A.
³ Technische Universität München Lehrstuhl für Mathematische Statistik 85748 Garching b. München Boltzmannstr. 3

Algebraic Combinatorics, Volume 6 (2023) no. 3, pp. 835-861.

Abstract

We investigate the class of Kazhdan–Lusztig varieties, and its subclass of matrix Schubert varieties, endowed with a naturally defined torus action. Writing a matrix Schubert variety $\bar{X_{w}}$ as $\bar{X_{w}} = Y_{w} \times ℂ^{d}$ (where $d$ is maximal possible), we show that $Y_{w}$ can be of complexity- $k$ exactly when $k \neq 1$ . Also, we give a combinatorial description of the extremal rays of the weight cone of a Kazhdan–Lusztig variety, which in particular turns out to be the edge cone of an acyclic directed graph. As a consequence we show that given permutations $v$ and $w$ , the complexity of Kazhdan–Lusztig variety indexed by $(v, w)$ is the same as the complexity of the Richardson variety indexed by $(v, w)$ . Finally, we use this description to compute the complexity of certain Kazhdan–Lusztig varieties.

Received: 2022-04-26
Revised: 2022-11-23
Accepted: 2022-11-23
Published online: 2023-06-19

DOI: 10.5802/alco.279

Classification: 14M15, 14M25, 52B20, 05E10, 05C20
Keywords: Schubert variety, Kazhdan–Lusztig variety, weight cone, torus action, toric variety, $T$-variety, edge cone, directed graph.

Author's affiliations:

Donten-Bury, Maria ¹; Escobar, Laura ²; Portakal, Irem ³

¹ University of Warsaw Institute of Mathematics Banacha 2 02-097 Warszawa Poland
² Department of Mathematics and Statistics Washington University in St. Louis One Brookings Drive St. Louis, Missouri 63130 U.S.A.
³ Technische Universität München Lehrstuhl für Mathematische Statistik 85748 Garching b. München Boltzmannstr. 3

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

@article{ALCO_2023__6_3_835_0,
     author = {Donten-Bury, Maria and Escobar, Laura and Portakal, Irem},
     title = {Complexity of the usual torus action on {Kazhdan{\textendash}Lusztig} varieties},
     journal = {Algebraic Combinatorics},
     pages = {835--861},
     publisher = {The Combinatorics Consortium},
     volume = {6},
     number = {3},
     year = {2023},
     doi = {10.5802/alco.279},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.279/}
}

TY  - JOUR
AU  - Donten-Bury, Maria
AU  - Escobar, Laura
AU  - Portakal, Irem
TI  - Complexity of the usual torus action on Kazhdan–Lusztig varieties
JO  - Algebraic Combinatorics
PY  - 2023
SP  - 835
EP  - 861
VL  - 6
IS  - 3
PB  - The Combinatorics Consortium
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.279/
DO  - 10.5802/alco.279
LA  - en
ID  - ALCO_2023__6_3_835_0
ER  -

%0 Journal Article
%A Donten-Bury, Maria
%A Escobar, Laura
%A Portakal, Irem
%T Complexity of the usual torus action on Kazhdan–Lusztig varieties
%J Algebraic Combinatorics
%D 2023
%P 835-861
%V 6
%N 3
%I The Combinatorics Consortium
%U https://alco.centre-mersenne.org/articles/10.5802/alco.279/
%R 10.5802/alco.279
%G en
%F ALCO_2023__6_3_835_0

Donten-Bury, Maria; Escobar, Laura; Portakal, Irem. Complexity of the usual torus action on Kazhdan–Lusztig varieties. Algebraic Combinatorics, Volume 6 (2023) no. 3, pp. 835-861. doi : 10.5802/alco.279. https://alco.centre-mersenne.org/articles/10.5802/alco.279/

References
Cited by

[1] Altmann, Klaus; Hausen, Jürgen Polyhedral divisors and algebraic torus actions, Math. Ann, Volume 334 (2006), pp. 557-607 | DOI | MR | Zbl

[2] Altmann, Klaus; Ilten, Nathan Owen; Petersen, Lars; Süß, Hendrik; Vollmert, Robert The geometry of T-varieties, Contributions to Algebraic Geometry. Impanga Lecture Notes. (Pragacz, Piotr, ed.), European Mathematical Society (EMS), 2012, pp. 17-69 | DOI | Zbl

[3] Altmann, Klaus; Wiśniewski, Jarosław A. P-divisors of Cox rings, Michigan Math. J., Volume 60 (2011), pp. 463-480 | MR | Zbl

[4] Billey, Sara; Lakshmibai, V. Singular loci of Schubert varieties, Progr. Math., 182, Birkhäuser Boston, Inc., Boston, MA, 2000, xii+251 pages | DOI | MR

[5] Björner, Anders; Brenti, Francesco Combinatorics of Coxeter groups, Grad. Texts in Math., 231, Springer, New York, 2005

[6] Brion, Michel Lectures on the Geometry of Flag Varieties, Topics in Cohomological Studies of Algebraic Varieties. Trends in Mathematics. (Pragacz, Piotr, ed.), Birkhäuser Basel, 2005, pp. 33-85 | DOI | Zbl

[7] Castillo, Federico; Cid-Ruiz, Yairon; Li, Binglin; Montaño, Jonathan; Zhang, Naizhen When are multidegrees positive?, Adv. Math., Volume 374 (2020), p. 107382, 34 pp | DOI | MR | Zbl

[8] Comtet, Louis Advanced combinatorics: the art of finite and infinite expansions; rev. version, Reidel, Dordrecht, 1974 (Trans. of : Analyse combinatoire. Paris : Presses Univ. de France, 1970) | DOI

[9] De Concini, C.; Lakshmibai, V. Arithmetic Cohen-Macaulayness and arithmetic normality for Schubert varieties, Amer. J. Math., Volume 103 (1981) no. 5, pp. 835-850 | DOI | MR | Zbl

[10] Escobar, Laura; Mészáros, Karola Toric Matrix Schubert Varieties and Their Polytopes, Proc. Amer. Math. Soc., Volume 144 (2016) no. 12, pp. 5081-5096 | DOI | MR | Zbl

[11] Fulton, William Flags, Schubert polynomials, degeneracy loci, and determinantal formulas, Duke Math. J., Volume 65 (1992) no. 3, pp. 381-420 | MR | Zbl

[12] Fulton, William Young tableaux, London Math. Soc. Stud. Texts, 35, Cambridge University Press, Cambridge, 1997, x+260 pages (With applications to representation theory and geometry) | MR

[13] Gitler, Isidoro; Reyes, Enrique; Villarreal, Rafael H. Ring graphs and complete intersections toric ideals, Duke Math. J., Volume 310 (2010), pp. 430-441 | MR | Zbl

[14] Insko, Erik; Yong, Alexander Patch ideals and Peterson varieties, Transform. Groups, Volume 17 (2012) no. 4, pp. 1011-1036 | DOI | MR | Zbl

[15] Karuppuchamy, Paramasamy On Schubert varieties, Comm. Algebra, Volume 41 (2013) no. 4, pp. 1365-1368 | DOI | MR | Zbl

[16] Kazhdan, David; Lusztig, George Representations of Coxeter Groups and Hecke Algebras, Invent. Math., Volume 53 (1979), pp. 165-184 | DOI | MR | Zbl

[17] Klein, Patricia; Weigandt, Anna Bumpless pipe dreams encode Gröbner geometry of Schubert polynomials, 2021 | arXiv

[18] Knutson, Allen; Miller, Ezra Gröbner geometry of Schubert polynomials, Ann. of Math., Volume 161 (2005), pp. 1245-1318 | DOI | Zbl

[19] Knutson, Allen; Woo, Alexander; Yong, Alexander Singularities of Richardson varieties, Math. Res. Lett., Volume 20 (2013) no. 2, pp. 391-400 | DOI | MR | Zbl

[20] Kodama, Yuji; Williams, Lauren The full Kostant-Toda hierarchy on the positive flag variety, Comm. Math. Phys., Volume 335 (2015) no. 1, pp. 247-283 | DOI | MR | Zbl

[21] Lee, Eunjeong; Masuda, Mikiya Generic torus orbit closures in Schubert varieties, J. Combin. Theory Ser. A, Volume 170 (2020), pp. 105-143 | DOI | MR | Zbl

[22] Lee, Eunjeong; Masuda, Mikiya; Park, Seonjeong On Schubert varieties of complexity one, Pacific J. Math., Volume 315 (2021) no. 2, pp. 419-447 | DOI | MR | Zbl

[23] Lee, Eunjeong; Masuda, Mikiya; Park, Seonjeong Toric Bruhat interval polytopes, J. Combin. Theory Ser. A, Volume 179 (2021), p. Paper No. 105387, 41 pp | DOI | MR | Zbl

[24] Ohsugi, Hidefumi; Hibi, Takayuki Toric Ideals Generated by Quadratic Binomials, J. Algebra, Volume 218 (1999) no. 2, pp. 509-527 | DOI | MR | Zbl

[25] Portakal, Irem On rigidity of toric varieties arising from bipartite graphs, J. Algebra, Volume 569 (2021), pp. 784-822 | DOI | MR | Zbl

[26] Portakal, Irem Rigid toric matrix Schubert varieties, J. Algebr. Comb. (2023) | DOI

[27] Ramanan, S.; Ramanathan, A. Projective normality of flag varieties and Schubert varieties, Invent. Math., Volume 79 (1985) no. 2, pp. 217-224 | DOI | MR | Zbl

[28] Setiabrata, Linus Faces of Root Polytopes, SIAM J. Discrete Math., Volume 35 (2021) no. 3, pp. 2093-2114 | DOI | MR | Zbl

[29] Tsukerman, E.; Williams, L. Bruhat interval polytopes, Adv. Math., Volume 285 (2015), pp. 766-810 | DOI | MR | Zbl

[30] Tymoczko, Julianna S. An introduction to equivariant cohomology and homology, following Goresky, Kottwitz, and MacPherson, Snowbird lectures in algebraic geometry (Contemp. Math.), Volume 388, Amer. Math. Soc., Providence, RI, 2005, pp. 169-188 | DOI | MR

[31] Woo, Alexander; Yong, Alexander Governing singularities of Schubert varieties, J. Algebra, Volume 320 (2008) no. 2, pp. 495-520 | DOI | MR | Zbl

[32] Woo, Alexander; Yong, Alexander A Gröbner basis for Kazhdan-Lusztig ideals, Amer. J. Math., Volume 134 (2012) no. 4, pp. 1089-1137 | Zbl

Cited by Sources: